How can I use a pivot circle instead of a pivot point
I think instead of trying to find a way of changing pivot point mechanism (which doesn't necessarily exist) it's better to change way of modeling this object.

Simple alternative is to model this using curves. Create 2 curve circles, one - for the main object, another one - as bevel object for the first one. Set parameters for beveling on the first one and then make proper sharpness of second one to achieve form of desired object.


Both curve circles next to each other. Handle type for the big one is left as default, for the small one (profile curve) is set to Vector (edit it by pressing V in Edit mode)


Options for the big circle object (main one)


By editing profile curve you change main one. Note that if you want length of edges outlined on your screenshot to remain the same make sure to scale profile vertices (selected above) simultaneously or length will become uneven.


Edges of the final mesh have the same length.

See related info here Extrude along path